In this chapter you will learn how to say precisely how long something is. With whole numbers only, we cannot always say precisely how long something is. Fractions were invented for that purpose. You will also learn to calculate with fractions.

6.1 Measuring accurately with parts of a unit 155

6.2 Different parts in different colours 160

6.3 Combining fractions 162

6.4 Tenths and hundredths (percentages) 164

6.5 Thousandths, hundredths and tenths 167

6.6 Fraction of a fraction 169

6.7 Multiplying with fractions 172

6.8 Ordering and comparing fractions 176

6 Fractions

6.1 Measuring accurately with parts of a unit

a strange measuring unit

In this activity, you will measure lengths with a unit called a greystick. The grey measuring stick below is exactly 1 greystick long. You will use this stick to measure different objects.

75007.png 

The red bar below is exactly 2 greysticks long.

74996.png 

As you can see, the yellow bar below is longer than 1 greystick but shorter than 2 greysticks.

74987.png 

To try to measure the yellow bar accurately, we will divide one greystick into six equal parts:

74979.png 

So each of these parts is one sixth of a greystick.

1. Do you think one can say the yellow bar is one and four sixths of a greystick long?


74944.png 

2. Describe the length of the blue bar in words.



This greystick ruler is divided into seven equal parts:

Each part is one seventh of a greystick.

74933.png

3. In each case below, say what the smaller parts of the greystick may be called. Write your answers in words.

74925.png

(a)


(b)


(c)


(d)


(e)


(f)


(g)


(h)


(i)


(j)


(k)


(l)


(m)


(n)


How did you find out what to call the small parts?




Write all your answers to the following questions in words.

4. (a) How long is the upper yellow bar?


74529.png 

(b) How long is the lower yellow bar?


5. (a) How long is the blue bar at the bottom of the previous page?


(b) How long is the red bar at the bottom of the previous page?



6. (a) How many twelfths of a greystick is the same length as one sixth of a greystick?


(b) How many twenty-fourths is the same length as one sixth of a greystick?


(c) How many twenty-fourths is the same length as seven twelfths of a greystick?



7. (a) How long is the upper yellow bar below?


74496.png 

(b) How long is the lower yellow bar above?


(c) How long is the blue bar?



(d) How long is the red bar?




8. (a) How many fifths of a greystick is the same length as 12 twentieths of a greystick?

74437.png

(b) How many fourths (or quarters) of a greystick is the same length as 15 twentieths of a greystick?


describe the same length in different ways

Two fractions may describe the same length. You can see here that three sixths of a greystick is the same as four eighths of a greystick.

74429.png

When two fractions describe the same portion we say they are equivalent.

1. (a) What can each small part on this greystick be called?


74388.png 

(b) How many eighteenths is one sixth of the greystick?


(c) How many eighteenths is one third of the greystick?


(d) How many eighteenths is five sixths of the greystick?


2. (a) Write (in words) the names of four different fractions that are all equivalent to three quarters. You may look at the yellow greysticks on page 154 to help you.



(b) Which equivalents for two thirds can you find on the yellow greysticks?



3. The information that 2 thirds is equivalent to 4 sixths, to 6 ninths and to 8 twelfths is written in the second row of the table below. Complete the other rows of the table in the same way. The diagrams on page 154 may help you.

thirds

fourths

fifths

sixths

eighths

ninths

tenths

twelfths

twentieths

1

2

-

-

4

-

6

-

8

-

-

3

-

-

1

-

-

2

-

-

3

-

-

4

4. Complete this table in the same way as the table in question 3.

fifths

tenths

fifteenths

twentieths

twenty-fifths

fiftieths

hundredths

1

2

3

4

5

6

7

5. Draw on the greysticks below to show that 3 fifths and 9 fifteenths are equivalent. Draw freehand; you need not measure and draw accurately.

74268.png 

6. Complete these tables in the same way as the table in question 4.

eighths

sixteenths

24ths

24ths

sixths

twelfths

18ths

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

7. (a) How much is five twelfths plus three twelfths?


(b) How much is five twelfths plus one quarter?


(c) How much is five twelfths plus three quarters?


(d) How much is one third plus one quarter? It may help if you work with the equivalent fractions in twelfths.


6.2 Different parts in different colours

This strip is divided into eight equal parts.

Five eighths of this strip is red.

74170.png

1. What part of the strip is blue?



2. What part of this strip is yellow?

74162.png


3. What part of the strip is red?



4. What part of this strip is coloured blue and what part is coloured red?

74149.png 



5. (a) What part of this strip is blue, what part is red and what part is white?

74139.png 


(b) Express your answer differently with equivalent fractions.



6. A certain strip is not shown here. Two ninths of the strip is blue, and three ninths of the strip is green. The rest of the strip is red. What part of the strip is red?



7. What part of the strip below is yellow, what part is blue, and what part is red?

74131.png 



The number of parts in a fraction is called the numerator of the fraction. For example, the numerator in 5 sixths is 5.

The type of part in a fraction is called the denominator. It is the name of the parts that are being referred to and it is determined by the size of the part. For example, sixths is the denominator in 5 sixths.

To enumerate means "to find the number of".

To denominate means "to give a name to".

/6and 74085.png are short ways to write sixths.

The numerator (number of parts) is written above

the line of the fraction: 74078.png 

The denominator is indicated by a number written

below the line: 74070.png

8. Consider the fraction 3 quarters. It can be written as 74053.png.

(a) Multiply both the numerator and the denominator by 2 to form a new fraction. Is the new fraction equivalent to 74047.png ? You may check on the diagram below.

74038.png

\times =


(b) Multiply both the numerator and the denominator by 3 to form a new fraction. Is the new fraction equivalent to 74013.png?

\times = Yes, it is equivalent to .

(c) Multiply both the numerator and the denominator by 4 to form a new fraction. Is the new fraction equivalent to 73981.png?

\times = Yes, it is equivalent to .

(d) Multiply both the numerator and the denominator by 6 to form a new fraction. Is the new fraction equivalent to 73946.png?

\times = Yes, it is equivalent to .

6.3 Combining fractions

bigger and smaller parts

Gertie was asked to solve this problem:

Ateam of road-builders built 73909.png km of road in one week, and 73902.png km in the next week. What is the total length of road that they built in the two weeks?

She thought like this to solve the problem:

73894.png is eight twelfths and 73887.png is ten twelfths, so altogether it is eighteen twelfths.

I can write 73878.png or "18 twelfths".

Ican also say 12 twelfths of a km is 1 km, so 18 twelfths is 1 km and 6 twelfths of a km.

This I can write as 1 73869.png. It is the same as 1 73861.png km.

Gertie was also asked the question: How much is 4 73854.png + 2 73845.png?

She thought like this to answer it:

4 73836.png is 4 wholes and 5 ninths, and 2 73828.png is 2 wholes and 7 ninths.

So altogether it is 6 wholes and 12 ninths. But 12 ninths is 9 ninths (1 whole) and 3 ninths, so I can say it is 7 wholes and 3 ninths.

I can write 7 73821.png.

1. Would Gertie be wrong if she said her answer was 7 73812.png?

No, 7 is equivalent to 7. ( is expressed in its simplest form.)

2. Senthereng has 4 73774.png bottles of cooking oil. He gives 1 73765.png bottles to his friend Willem. How much oil does Senthereng have left?

3 or 3

3. Margaret has 5 73738.png bottles of cooking oil. She gives 3 73731.png bottles to her friend Naledi. How much oil does Margaret have left?

5 - 3 = 4 - 3 = 1 or 1

4. Calculate each of the following:

(a) 4 73676.png - 3 73668.png (b) 3 73660.png + 73649.png

= 3 76386.jpg - 3 76377.jpg 

= 3 76396.jpg 

= 76361.jpg 

= 4 76369.jpg 

(c) 3 73545.png + 1 88146.png (d) 4 88165.png - 2 88157.png 

= 4 76336.jpg 

= 3 76353.jpg - 3 76344.jpg 

= 5 76312.jpg 

= 76329.jpg or 76320.jpg 

(e) 1 73412.png - 73403.png (f) 3 73396.png - 1 73388.png

= 1 76289.jpg - 76285.jpg = 76278.jpg - 76270.jpg 

= 3 76308.jpg - 1 76298.jpg 

= 76261.jpg 

= 2

(g) 73287.png + 73282.png + 73278.png + 73274.png + 73268.png (h) 6 73258.png + 2 73251.png - 73241.png

= 76228.jpg 

= 6 76254.jpg + 2 76246.jpg - 76236.jpg

= 3 76210.jpg 

= 8 76220.jpg 

(i) 73139.png + 73135.png + 73131.png + 73127.png + 73121.png + 73117.png + 73113.png + 73106.png + 73101.png + 73097.png + 73091.png + 73085.png + 73076.png

= 73069.jpg = 8 73061.jpg 

(j) 2 73056.png + 2 73052.png + 2 73048.png + 2 73041.png + 2 73036.png + 2 73032.png + 2 73024.png + 2 73016.png

= 16 73009.jpg = 20 73001.jpg 

(k) (4 72994.png + 1 72985.png) - 2 72977.png 

= 5 72970.jpg - 2 72958.jpg = 5 72949.jpg - 2 72942.jpg = 3 72933.jpg 

(l) (2 72928.png + 3 72924.png) - (1 72916.png + 3 72909.png)

= ( 72901.jpg - 72894.jpg) - ( 72884.jpg + 72873.jpg) = - 72865.jpg = - 5 72857.jpg or - 5 72846.jpg

5. Neo's report had five chapters. The first chapter was 72840.png of a page, the second chapter was 2 72836.png pages, the third chapter was 3 72829.png pages, the fourth chapter was 3 pages and the fifth chapter was 1 72822.png pages. How many pages was Neo's report in total?

72817.jpg + 2 72813.jpg + 3 72805.jpg + 3 + 1 72799.jpg = 9 72791.jpg = 11 72784.jpg pages or 11 72775.jpg pages

6.4 Tenths and hundredths (percentages)

1. (a) 100 children each get 3 biscuits. How many biscuits is this in total?


(b) 500 sweets are shared equally between 100 children. How many sweets does each child get?



2. The picture below shows a strip of licorice. The very small pieces can easily be broken off on the thin lines. How many very small pieces are shown on the picture?

72764.png 



3. Gatsha runs a spaza shop. He sells strips of licorice like the above for R2 each.

(a) What is the cost of one very small piece of licorice, when you buy from Gatsha?


(b) Jonathan wants to buy one fifth of a strip of licorice. How much should he pay?


(c) Batseba eats 25 very small pieces. What part of a whole strip of licorice is this?

or

Each small piece of the above strip is one hundredth of the whole strip.

4. (a) Why can each small piece be called one hundredth of the whole strip?


(b) How many hundredths is the same as one tenth of the strip?



Gatsha often sells parts of licorice strips to customers. He uses a "quarters marker" and a "fifths marker" to cut off the pieces correctly from full strips. His two markers are shown below, next to a full strip of licorice.

72740.png 

5. (a) How many hundredths is the same as two fifths of the whole strip?


(b) How many tenths is the same as 72731.png of the whole strip?


(c) How many hundredths is the same as 72723.png of the whole strip?


(d) Freddie bought 72716.png of a strip. How many fifths of a strip is this?


(e) Jamey bought part of a strip for R1,60. What part of a strip did she buy?


6. Gatsha, the owner of the spaza shop, sold pieces of yellow licorice to different children. Their pieces are shown below. How much (what part of a whole strip) did each of them get?

72706.png 



7. The yellow licorice shown above costs R2,40 (240 cents) for a strip. How much does each of the children have to pay? Round off the amounts to the nearest cent.



8. (a) How much is 72698.png of 300 cents? (b) How much is 72690.png of 300 cents?





(c) How much is 72632.png of 300 cents? (d) How much is 72623.png of 300 cents?





(e) How much is 72566.png of 300 cents? (f) How much is 72559.png of 300 cents?

40 \times 3c = 120c or R1,20

76202.jpg of 300c = 60c so 76195.jpg = 120c or R1,20

9. Explain why your answers for questions 8(e) and 8(f) are the same.

Because and are equivalent fractions

Another word for hundredth is per cent.

Instead of saying

we can say

The symbol for per cent is %.

10. How much is 80% of each of the following?

(a) R500 (b) R480 (c) R850 (d) R2 400



11. How much is 8% of each of the amounts in 10(a), (b), (c) and (d)?



12. How much is 15% of each of the amounts in 10(a), (b), (c) and (d)?



13. Building costs of houses increased by 20%. What is the new building cost for a house that previously cost R110 000 to build?



14. The value of a car decreases by 30% after one year. If the price of a new car is R125 000, what is the value of the car after one year?


15. Investigate which denominators of fractions can easily be converted to powers of 10.


6.5 Thousandths, hundredths and tenths

many equal parts

1. In a camp for refugees, 50 kg of sugar must be shared equally between 1 000 refugees. How much sugar will each refugee get? Keep in mind that 1 kg is 1 000 g. You can give your answer in grams.


2. How much is each of the following?

(a) one tenth of R6 000 (b) one hundredth of R6 000





(c) one thousandth of R6 000 (d) ten hundredths of R6 000





(e) 100 thousandths of R6 000 (f) seven hundredths of R6 000





(g) 70 thousandths of R6 000 (h) seven thousandths of R6 000





3. Calculate.

(a) 72283.png + 72276.png (b) 3 72266.png + 2 72258.png 

= 76163.jpg + 76155.jpg 

= 3 76187.jpg + 2 76179.jpg = 5 76170.jpg 

= 76135.jpg or 76128.jpg

= 6 76144.jpg 

(c) 72155.png + 72145.png (d) 72140.png + 72133.png 

= 76103.jpg + 76096.jpg 

= 76119.jpg or 76110.jpg 

= 76087.jpg 

= 1

(e) 72033.png + 72023.png (f) 72014.png + 72007.png

= 76049.jpg + 76041.jpg 

= 76077.jpg + 76070.jpg = 76061.jpg 

= 76024.jpg 

= 76034.jpg 

4. Calculate.

(a) 71903.png + 71893.png + 71885.png (b) 71877.png + 71868.png + 71859.png

= 75987.jpg + 75979.jpg + 75971.jpg 

= 76018.jpg + 76011.jpg + 76003.jpg = 75996.jpg 

= 75944.jpg or 75935.jpg 

= 1 75961.jpg or 1 75952.jpg 

(c) 71755.png + 71748.png + 71739.png (d) 71729.png + 71721.png + 71712.png

= 75906.jpg + 75897.jpg + 75888.jpg = 75881.jpg

= 75927.jpg + 75920.jpg + 75913.jpg 

= 75854.jpg = 1 75845.jpg or 1 75837.jpg 

= 75871.jpg or 75862.jpg 

5. In each case investigate whether the statement is true or not, and give reasons for your final decision.

(a) 71610.png + 71603.png + 71598.png = 71592.png + 71583.png + 71575.png

71568.jpg + 71561.jpg + 71552.jpg = 71545.jpg 71538.jpg + 71528.jpg + 71519.jpg = 71511.jpg 

True, because LHS equals RHS.

(b) 71505.png + 71499.png + 71495.png = 71487.png + 71479.png + 71474.png

LHS is the same as in the previous question but RHS is 71467.jpg.

Not true.

(c) 71460.png + 71452.png + 71444.png = 71439.png + 71433.png + 71427.png

True, because LHS is again 71420.jpg and the sum on the RHS forms the thousandths

digits one by one.

(d) 71413.png = 71404.png + 71395.png + 71387.png


Both sides can be expressed as 71379.jpg + 71369.jpg + 71361.jpg.

6.6 Fraction of a fraction

form parts of parts

1. (a) How much is 1 fifth of R60?


(b) How much is 3 fifths of R60?


2. How much is 7 tenths of R80? (You may first work out how much 1 tenth of R80 is.)


3. In the USA the unit of currency is the US dollar, in Britain it is the pound, in Western Europe the euro, and in Botswana the pula.

(a) How much is 2 fifths of 20 pula?


(b) How much is 2 fifths of 20 euro?


(c) How much is 2 fifths of 12 pula?


4. Why was it so easy to calculate 2 fifths of 20, but difficult to calculate 2 fifths of 12?


There is a way to make it easy to calculate something like 3 fifths of R4. You just change the rands to cents!

5. Calculate each of the following. You may change the rands to cents to make it easier.

(a) 3 eighths of R2,40 (b) 7 twelfths of R6





(c) 2 fifths of R21 (d) 5 sixths of R3





6. You will now do some calculations about secret objects.

(a) How much is 3 tenths of 40 secret objects?


(b) How much is 3 eighths of 40 secret objects?


The secret objects in question 6 are fiftieths of a rand.

7. (a) How many fiftieths is 3 tenths of 40 fiftieths?


(b) How many fiftieths is 5 eighths of 40 fiftieths?


8. (a) How many twentieths of a kilogram is the same as 71139.png of a kilogram?


(b) How much is one fifth of 15 rands?


(c) How much is one fifth of 15 twentieths of a kilogram?


(d) So, how much is one fifth of 71057.png of a kilogram?


9. (a) How much is 71024.png of 24 fortieths of some secret object?


(b) How much is 70993.png of 24 fortieths of the secret object?


10. Do you agree that the answers for the previous question are 2 fortieths and 14 fortieths? If you disagree, explain why you disagree.


11. (a) How much is 70962.png of 80?


(b) How much is 70934.png of 80?


(c) How much is 70902.png of 80?


(d) How much is 70871.png of 80?


(e) Explain why 70838.png of 80 is the same as 70829.png of 80.

Because and are equivalent fractions

12. Look again at your answers for questions 9(b) and 11(e). How much is 70803.png of 70793.png?Explain your answer.

It is because = [from question 11(e)].

So of is the same as of , which was calculated in question 9(b) as .

The secret object in question 9 was an envelope with R160 in it.

After the work you did in questions 9, 10 and 11, you know that

It is easy to calculate 70704.png of 70696.png: 1 twelfth of 24 is 2, so 7 twelfths of 24 is 14, so

7twelfths of 24 fortieths is 14 fortieths.

70692.png of 70686.png can be calculated in the same way. But 1 eighth of 70681.png is a slight problem, so it would be better to use some equivalent of 70677.png. The equivalent should be chosen so that it is easy to calculate 1 eighth of it; so it would be nice if the numerator could be 8.

70670.png is equivalent to 70665.png, so instead of calculating 70661.png of 70657.png we may calculate 70652.png of 70647.png.

13. (a) Calculate 70642.png of 70634.png .

= of is of is

(b) So, how much is 70559.png of 70550.png ?

= so the answer is the same as in question 13(a), that is .

14. In each case replace the second fraction by a suitable equivalent, and then calculate.

(a) How much is 70515.png of 70504.png?

= of =

(b) How much is 70460.png of 70452.png?

= of =

(c) How much is 70399.png of 70392.png?

= of =

(d) How much is 70346.png of 70339.png?

= of =

6.7 Multiplying with fractions

parts of rectangles, and parts of parts

1. (a) Divide the rectangle on the left into eighths by drawing vertical lines. Lightly shade the left 3 eighths of the rectangle.

(b) Divide the rectangle on the right into fifths drawing horizontal lines. Lightly shade the upper 2 fifths of the rectangle.

70293.png 

2. (a) Shade 4 sevenths of the rectangle on the left below.

(b) Shade 16 twenty-eighths of the rectangle on the right below.

70284.png 

3. (a) What part of each big rectangle below is coloured yellow?


(b) What part of the yellow part of the rectangle on the right is dotted?


70227.png 

(c) Into how many squares is the whole rectangle on the right divided?


(d) What part of the whole rectangle on the right is yellow and dotted?



4. Make diagrams on the grid below to help you to figure out how much each of the following is:

(a) 70198.png of 70191.png (b) 70184.png of 70175.png 

= 75823.jpg 

= 75830.jpg 

70117.png 

Here is something you can do with the fractions 70113.png and 70109.png:

Multiply the two numerators and make this the numerator of a new fraction. Also multiply the two denominators, and make this the denominator of a new fraction 70105.png = 70096.png.

5. Compare the above with what you did in question 14(a) of section 6.6 and in

question 4(a) at the top of this page. What do you notice about 70087.png of 70078.png and 70068.png?

The answers are the same, so multiplying the numerators with each other and

the denominators with each other seems to be a way of finding the answer to of .

6. (a) Alan has 5 heaps of 8 apples each. How many apples is that in total?


(b) Sean has 10 heaps of 6 quarter apples each. How many apples is that in total?

10 \times = = 15 apples

Instead of saying 70032.png of R40 or 70028.png of 70023.png of a floor surface, we may say 70017.png \times R40 or 70008.png \times 70001.png of a floor surface.

7. Use the diagrams below to figure out how much each of the following is:

(a) 69983.png \times 69973.png (b) 69966.png \times 69956.png 

= 75806.jpg 

= 75815.jpg 

69897.png

8. (a) Perform the calculations 69893.png for 69886.png and 69879.png and compare the answer to your answer for question 7(a).

= The answers are the same.

(b) Do the same for 69856.png and 69848.png.

= The answers are the same.

9. Perform the calculations 69823.png for

(a) 69816.png and 69812.png (b) 69808.png and 69804.png

= 75790.jpg 

= 75797.jpg 

10. Use the diagrams below to check whether the formula 69748.pngproduces the correct answers for 69738.png \times 69730.png and 69722.png \times 69714.png .



69705.png

11. Calculate each of the following:

(a) 87631.png of 87622.png of R60 (b) 87614.png of 87607.png of R63 (c) 87599.png of 87589.png of R45







12. (a) John normally practises soccer for three quarters of an hour every day. Today he practised for only half his usual time. How long did he practise today?

\times = of an hour

\times 60 minutes = = 22 or 22 minutes

(b) A bag of peanuts weighs 87463.png of a kilogram. What does 87456.png of a bag weigh?

\times = kg

(c) Calculate the mass of 7 87430.png packets of sugar if 1 packet has a mass of 87423.pngkg.

7 \times + \times or \times

= + = + =

= = 5kg = 5kg

87327.png 

6.8 Ordering and comparing fractions

1. Order the following from the smallest to the biggest:

(a) 87318.png; 87308.png; 87301.png; 87294.png; 87284.png (b) 87275.png; 87265.png; 87257.png; 73%; 87249.png

= 87686.jpg 

= 87690.jpg 

= 87674.jpg 

= 87680.jpg 

2. Order the following from the biggest to the smallest:

(a) 87144.png; 87137.png; 87129.png; 87119.png; 87112.png (b) 87105.png; 87096.png; 87087.png; 87078.png; 87068.png

= 87664.jpg 

= 87668.jpg 

= 87649.jpg 

= 87656.jpg 

3. Use the symbols = , > or < to make the following true:

(a) 86964.png

86931.png (b) 86924.png

86892.png 

1. Do the calculations given below. Rewrite each question in the common fraction notation. Then write the answer in words and in the common fraction notation.

(a) 3 twentieths + 5 twentieths (b) 5 twelfths + 11 twelfths

75694.jpg + 75685.jpg = 75677.jpg

75721.jpg + 75714.jpg = 75706.jpg

Eight twentieths

Sixteen twelfths

(c) 3 halves + 5 quarters (d) 3 fifths + 3 tenths

75630.jpg + 75626.jpg = 75619.jpg + 75610.jpg = 75600.jpg = 2 75591.jpg 

75666.jpg + 75661.jpg = 75654.jpg + 75647.jpg = 75639.jpg 

Two wholes and three quarters

Nine tenths

2. Complete the equivalent fractions.

(a) 69072.png =

77251.png
77258.png

(b) 69052.png =

77261.png
77268.png

(c) 69035.png =

77278.png
77271.png

(d) 69015.png =

77281.png
77288.png

(e) 68997.png =

77301.png
77294.png

(f) 68979.png =

77311.png
77304.png

3. Do the calculations given below. Rewrite each question in words. Then write the answer in words and in the common fraction notation.

(a) 68961.png + 68952.png (b) 68943.png + 68934.png 

3 tenths plus 7 thirtieths

2 fifths plus 7 twelfths

= 9 thirtieths + 7 thirtieths

= 24 sixtieths + 35 sixtieths

= 16 thirtieths = 75573.jpg 

= 59 sixtieths = 75582.jpg 

(c) 68779.png + 68771.png (d) 86405.png - 86398.png 

1 hundredth plus 7 tenths

5 eighths minus 1 third

= 1 hundredth + 70 hundredths

=15 twenty-fourths - 8 twenty-fourths

= 71 hundredths = 75555.jpg 

= 7 twenty-fourths = 75563.jpg 

(e) 2 68601.png + 5 68593.png

2 wholes and 3 tenths plus 5 wholes and 9 tenths

= 7 wholes and 12 tenths

= 8 wholes and 2 tenths = 8 68585.jpg

4. Joe earns R5 000 per month. His salary increases by 12%. What is his new salary?

\times 5 000 = = 600 R5 000 + R600 = R5 600 per month

5. Ahmed earned R7 500 per month. At the end of a certain month, his employer raised his salary by 10%. However, one month later his employer had to decrease his salary again by 10%. What was Ahmed's salary then?

Increased by 10%: R7 500 + R750 = R8 250 per month

Decreased by 10%: R8 250 - R825 = R7 425 per month

6. Calculate each of the following and simplify the answer to its lowest form:

(a) 68556.png - 68549.png (b) 3 68540.png - 1 68532.png 

= 75516.jpg - 75507.jpg = 75499.jpg = 75490.jpg 

= 3 75548.jpg - 1 75540.jpg = 2 75532.jpg = 2 75525.jpg 

(c) 5 68477.png - 2 68469.png (d) 68465.png + 68456.png 

= 5 75450.jpg - 2 75443.jpg = 3 75436.jpg 

= 75481.jpg + 75472.jpg = 75465.jpg = 1 75458.jpg 

7. Evaluate.

(a) 68400.png \times 9 (b) 68391.png \times 68380.png (c) 68374.png \times 15 (d) 68370.png \times 68360.png 

= 75386.jpg = 4 75378.jpg 

= 75403.jpg = 75394.jpg 

= 75413.jpg = 10

= 75427.jpg = 75422.jpg 

8. Calculate.

(a) 2 68259.png \times 2 68252.png (b) 8 68245.png \times 3 68238.png 

= 75351.jpg \times 75342.jpg = 75334.jpg = 7 75326.jpg 

= 75371.jpg \times 75362.jpg = 75355.jpg = 28

(c) ( 68182.png + 68175.png) \times 68167.png (d) 68157.png \times 68149.png \times 68140.png 

= ( 75298.jpg + 75290.jpg) \times 75282.jpg = 75274.jpg \times 75267.jpg = 75258.jpg = 75249.jpg 

= 75316.jpg = 75305.jpg 

(e) 68081.png + 68072.png \times 68064.png (f) 68057.png - 68050.png \times 68042.png

= 75197.jpg + 75189.jpg = 75181.jpg = 75174.jpg 

= 75241.jpg - 75234.jpg = 75222.jpg - 75213.jpg = 75206.jpg 

Maths_English_TG_Grade%207_Book%20A_Chapter%206_final-1.tif
Maths_English_TG_Grade%207_Book%20A_Chapter%206_final-2.tif

In this chapter you will learn more about decimal fractions and how they relate to common fractions and percentages. You will also learn to order and compare decimal fractions, and how to calculate with decimal fractions.

7.1 Other symbols for tenths and hundredths 181

7.2 Percentages and decimal fractions 183

7.3 Decimal measurements 186

7.4 More decimal concepts 188

7.5 Ordering and comparing decimal fractions 190

7.6 Rounding off 192

7.7 Addition and subtraction with decimal fractions 193

7.8 Multiplication and decimal fractions 195

7.9 Division and decimal fractions 199

Maths%20Grade%207AS%20chapter%207%20frontispiece-1.tif

8 tenths

7 hundredths

5 thousandths

7 eighths

7 The decimal notation for fractions

7.1 Other symbols for tenths and hundredths

tenths and hundredths again …

1. (a) What part of the rectangle below is coloured yellow?

10 hundredths or or 1 tenth or

82123.png 

(b) What part of the rectangle is red? What part is blue? What part is green, and what part is not coloured?

Red: Blue: / Green: Not coloured:

0,1 is another way to write 82079.png and

82056.png is called the (common) fraction notation

and 0,1 is called the decimal notation.

2. Write the answers for 1(a) and (b) in decimal notation.



3. 3 tenths and 7 hundredths of a rectangle is coloured red, and 2 tenths and 6 hundredths of the rectangle is coloured brown. What part of the rectangle (how many tenths and how many hundredths) is not coloured? Write your answer in fraction notation and in decimal notation.

3 tenths () and 7 hundredths () is not coloured. That is or 0,37.

4. On Monday, Steve ate 3 tenths and 7 hundredths of a strip of licorice. On Tuesday, Steve ate 2 tenths and 5 hundredths of a strip of licorice. How much licorice did he eat on Monday and Tuesday together? Write your answer in fraction notation and in decimal notation.

0,62

5. Lebogang's answer for question 4 is 5tenths and 12 hundredths. Susan's answer is 6 tenths and 2 hundredths. Who is right, or are they both wrong?



The same quantity can be expressed in different ways in tenths and hundredths.

6. What is the decimal notation for each of the following numbers?

(a) 3 81997.png (b) 4 81989.png (c) 81981.png (d) 81974.png 









… and thousandths

0,001 is another way of writing 81871.png.

1. What is the decimal notation for each of the following?

(a) 81851.png (b) 81843.png (c) 81835.png (d) 81827.png









2. Write the following numbers in the decimal notation:

(a) 2 + 81724.png + 81717.png + 81712.png (b) 12 + 81704.png + 81699.png





(c) 2 + 81642.png (d) 67 81634.png





(e) 34 81577.png (f) 654 81570.png 





7.2 Percentages and decimal fractions

hundredths, percentages and decimals

1. The rectangle below is divided into small parts.

81510.png 

(a) How many of these small parts are there in the rectangle? And in one tenth of the rectangle?


(b) What part of the rectangle is blue? What part is green? What part is red?

Blue: / Green: / Red:

Instead of 6hundredths, you may say 6 per cent. It means the same.

2. Use the word "per cent" to say what part of the rectangle is green. What part is red?



3. What percentage of the rectangle is blue? What percentage is white?

We do not say: "How many per cent of the rectangle is green?"

We say: "What percentage of the rectangle is green?"



The symbol % is used for "per cent".

Instead of writing "17 per cent", you may write 17%.

Per cent means hundredths. The symbol % is a bit like the symbol 81427.png.

4. (a) How much is 1% of R400? (In other words: How much is 81409.png or 0,01 of R400?)


(b) How much is 37% of R400?


(c) How much is 37% of R700?


5. (a) 25 apples are shared equally between 100 people. How much apple does each person get? Write your answer as a common fraction and as a decimal fraction.

or 0,25

(b) How much is 1% (one hundredth) of 25?

or or 0,25

(c) How much is 8% of 25?


(d) How much is 8% of 50? And how much is 0,08 of 50?



0,37 and 37% and 81376.png are different symbols for the

same thing: 37 hundredths.

6. Express each of the following in three ways:

(a) 3 tenths (b) 7 hundredths

0,3 30% 82483.jpg 

0,07 7% 82491.jpg 

(c) 37 hundredths (d) 7 tenths

0,37 37% 82468.jpg 

0,7 70% 82475.jpg 

(e) 3 quarters (f) 7 eighths

0,75 75% 82459.jpg 

0,875 87,5% as hundredths, not possible

7. (a) How much is 3 tenths of R200 and 7 hundredths of R200 altogether?


(b) How much is 81212.png of R200?


(c) How much is 0,37 of R200?


(d) And how much is 37% of R200?


8. Express each of the following in three ways:

(a) 20 hundredths (b) 50 hundredths

0,2 20% 82441.jpg 

0,5 50% 82451.jpg 

(c) 25 hundredths (d) 75 hundredths

0,25 25% 82428.jpg 

0,75 75% 82436.jpg 

9. (a) Jan eats a quarter of a watermelon. What percentage of the watermelon is this?


(b) Sibu drinks 75% of the milk in a bottle. What fraction of the milk is this?

or

(c) Jeminah uses 0,75 (7 tenths and 5 hundredths) of the paint in a tin. What percentage of the paint does she use?



10. The floor of a large room is shown alongside.What part of the floor is covered in each of the four colours? Express your answer in four ways:

81089.png

(a) in the common fraction notation, usinghundredths,

(b) in the decimal notation,

(c) in the percentage notation, and

(d) if possible, in the common fraction notation,

as tenths and hundredths (for example 81080.png + 81073.png).

(a)

(b)

(c)

(d)

white

red

yellow

black

7.3 Decimal measurements

measuring on a number line

1. Read the lengths at the marked points (A to D) on the number lines. Give your answers as accurate as possible in decimal notation.

(a)

80976.png 

0,2 0,7 1,6 1,85

(b)

80968.png 

0,2 0,7 1,6 1,85

(c)

80960.png 

6,9 7,2 8,4 8,75

(d)

80948.png 

3,09 3,14 3,19 3,265

(e)

80939.png 

2,461 2,463 2,466 2,4685

(f)

80930.png 

0,4499 0,4502 0,4505 0,4509

(g)

80920.png 

10,4 11,2 12,4 13,4

2. Show the following numbers on the number line below:

(a) 0,6 (b) 1,2 (c) 1,85 (d) 2,3

(e) 2,65 (f) 3,05 (g) 0,08

80910.png 

3. Show the following numbers on the number line below:

(a) 3,06 (b) 3,08 (c) 3,015

(d) 3,047 (e) 3,005

80900.png

7.4 More decimal concepts

decimal jumps

Write the next ten numbers in the number sequences and show your number sequences, as far as possible, on the number lines.

1. (a) 0,2; 0,4; 0,6;


(b)

80865.png 

(c) How many 0,2s are there in 1?


(d) Write 0,2 as a common fraction.

82421.jpg 

2. (a) 0,3; 0,6; 0,9;


(b)

80784.png 

(c) How many 0,3s are there in 3?


(d) Write 0,3 as a common fraction.

82412.jpg 

3. (a) 0,25; 0,5;


(b)

80700.png 

(c) How many 0,25s are there in 1?


(d) Write 0,25 as a common fraction.

82403.jpg 

Acalculator can be programmed to do the same operation over and over again.

4. You can check your answers for questions 1 to 3 with a calculator. Program the calculator to help you.

5. Write the next five numbers in the number sequences:

(a) 9,3; 9,2; 9,1;


(b) 0,15; 0,14; 0,13; 0,12;


6. Check your answers with a calculator. Program the calculator to help you.

place value

1. Write each of the following as one number:

(a) 2 + 0,5 + 0,07 (b) 2 + 0,5 + 0,007





(c) 2 + 0,05 + 0,007 (d) 5 + 0,4 + 0,03 + 0,001





(e) 5 + 0,04 + 0,003 + 0,1 (f) 5 + 0,004 + 0,3 + 0,01





We can write 3,784 in expanded notation as 3,784 = 3 + 0,7 + 0,08 + 0,004.We can also name these parts as follows:

We say: the value of the 7 is 7 tenths but the place value of the 7 is tenths, because any digit in that place will represent the number of tenths.

2. Now write the value (in decimal fractions) and the place value of each of the underlined digits.

(a) 2,345 (b) 4,678 (c) 1,953







(d) 34,856 (e) 564,34 (f) 0,987







7.5 Ordering and comparing decimal fractions

from biggest to smallest and smallest to biggest

1. Order the following numbers from biggest to smallest. Explain your method.

0,8 0,05 0,5 0,15 0,465 0,55 0,75 0,4 0,62





2. Below are the results of some of the 2012 London Olympic events. In each case, order them from first to last place. Use the column provided.

(a) Women: Long jump – Final

Name

Country

Distance

Place

Anna Nazarova

RUS

6,77 m

Brittney Reese

USA

7,12 m

Elena Sokolova

RUS

7,07 m

Ineta Radevica

LAT

6,88 m

Janay DeLoach

USA

6,89 m

3rd

Lyudmila Kolchanova

RUS

6,76 m

(b) Women: 400 m hurdles – Final

Name

Country

Time

Place

Georganne Moline

USA

53,92 s

Kaliese Spencer

JAM

53,66 s

4th

Lashinda Demus

USA

52,77 s

Natalya Antyukh

RUS

52,70 s

T'erea Brown

USA

55,07 s

Zuzana Hejnová

CZE

53,38 s

(c) Men: 110 m hurdles – Final

Name

Country

Time

Place

Aries Merritt

USA

12,92 s

Hansle Parchment

JAM

13,12 s

Jason Richardson

USA

13,04 s

Lawrence Clarke

GBR

13,39 s

Orlando Ortega

CUB

13,43 s

Ryan Brathwaite

BAR

13,40 s

(d) Men: Javelin – Final

Name

Country

Distance

Place

Andreas Thorkildsen

NOR

82,63 m

Antti Ruuskanen

FIN

84,12 m

Keshorn Walcott

TRI

84,58 m

Oleksandr Pyatnytsya

UKR

84,51 m

Tero Pitkämäki

FIN

82,80 m

Vítezslav Veselý

CZE

83,34 m

3. In each case, give a number that falls between the two numbers.

(This means you may give any number that falls anywhere between the two numbers.)

(a) 3,5 and 3,7 (b) 3,9 and 3,11 (c) 3,1 and 3,2







4. How many numbers are there between 3,1 and 3,2?


5. Fill in <, > or =.

(a) 0,4

0,52 (b) 0,4

0,32

(c) 2,61

2,7 (d) 2,4

2,40

(e) 2,34

2,567 (f) 2,34

2,251

7.6 Rounding off

Just as whole numbers can be rounded off to the nearest 10, 100 or 1 000, decimal fractions can be rounded off to the nearest whole number or to one, two, three etc. digits after the comma. A decimal fraction is rounded off to the number whose value is closest to it. Therefore 13,24 rounded off to one decimal place is 13,2 and 13,26 rounded off to one decimal place is 13,3. A decimal ending in a 5 is an equal distance from the two numbers to which it can be rounded off. Such decimals are rounded off to the biggest number, so 13,15 rounded off to one decimal place becomes 13,2.

saying it nearly but not quite

1. Round each of the following numbers off to the nearest whole number:

7,6 18,3 204,5 1,89 0,9 34,7 11,5 0,65



2. Round each of the following numbers off to one decimal place:

7,68 18,93 21,47 0,643 0,938 1,44 3,81



3. Round each of the following numbers off to two decimal places:

3,432 54,117 4,809 3,762 4,258 10,222 9,365 299,996



round off to help you calculate

1. John and three of his brothers sell an old bicycle for R44,65. How can the brothers share the money fairly?




2. A man buys 3,75 m of wood at R11,99 per metre. What does the wood cost him?



3. Estimate the answers of each of the following by rounding off the numbers:

(a) 89,3 \times 3,8 (b) 227,3 + 71,8 - 28,6





7.7 Addition and subtraction with decimal fractions

mental calculations

1. Complete the number chain.

34,123



+ 20



54,123





454,123





454,023





422,011





452,011





452,021





452,023





222,011





222,211





222,231





222,232





222,489





222,482





222,422





222,222

When you add or subtract decimal fractions, you can change them to common fractions to make the calculation easier.

2. Calculate each of the following:

(a) 0,7 + 0,2 (b) 0,7 + 0,4 (c) 1,3 + 0,8







(d) 1,35 + 0,8 (e) 0,25 + 0,7 (f) 0,25 + 0,07







(g) 3 - 0,1 (h) 3 - 0,01 (i) 2,4 - 0,5







some real-life problems

1. The owner of an internet café looks at her bank statement at the end of the day. She finds the following amounts paid into her account: R281,45; R39,81; R104,54 and R9,80. How much money was paid into her account on that day?



2. At the beginning of a journey the odometer in a car reads: 21589,4. At the end of the journey the odometer reads: 21763,7. What distance was covered?



3. At an athletics competition, an athlete runs the 100 m race in 12,8 seconds. The announcer says that the athlete has broken the previous record by 0,4 seconds. What was the previous record?



4. In a surfing competition five judges give each contestant a mark out of 10. The highest and the lowest marks are ignored and the other three marks are totalled. Work out each contestant's score and place the contestants in order from first to last.

A: 7,5 8 7 8,5 7,7 B: 8,5 8,5 9,1 8,9 8,7

C: 7,9 8,1 8,1 7,8 7,8 D: 8,9 8,7 9 9,3 9,1





5. A pipe is measured accurately. AC = 14,80 mm and

AB = 13,97 mm.

79193.png

How thick is the pipe (BC)?




6. Mrs Mdlankomo buys three packets of mincemeat. The packets weigh 0,356 kg, 1,201 kg and 0,978 kg respectively. What do they weigh together?



7.8 Multiplication and decimal fractions

the power of ten

1. (a) Complete the multiplication table.

\times

1 000

100

10

1

0,1

0,01

0,001

6

6 000

60

0,06

6,4

640

0,5

0,05

4,78

4 780

47,8

41,2

41 200

(b) Is it correct to say that "multiplication makes bigger"? When does multiplication make bigger?


(c) Formulate rules for multiplying with 10; 100; 1 000; 0,1; 0,01 and 0,001. Can you explain the rules?





(d) Now use your rules to calculate each of the following:

0,5 \times 10 0,3 \times 100 0,42 \times 10 0,675 \times 100



2. (a) Complete the division table.

\div

0,001

0,01

0,1

1

10

100

1 000

6

6

0,6

0,06

6,4

64

6,4

0,5

0,005

4,78

47,8

41,2

4 120

(b) Is it correct to say that "division makes smaller"? When does division make smaller?



(c) Formulate rules for dividing with 10; 100; 1 000; 0,1; 0,01 and 0,001. Can you explain the rules?





(d) Now use your rules to calculate each of the following:

0,5 \div 10 0,3 \div 100 0,42 \div 10


3. Complete the following:

(a) Multiplying with 0,1 is the same as dividing by


(b) Dividing by 0,1 is the same as multiplying by


Now discuss it with a partner or explain to him or her why this is so.

4. Fill in the missing numbers:

1,23456



\times 10



12,3456





123,456





12 345,6





1 234,56





12 345,6





123 456





1 234 560





123,456





1,23456





0,123456





123 456

78849.png 

What does multiplying a decimal number with a whole number mean?

What does something like 4 \times 0,5 mean?

What does something like 0,5 \times 4 mean?

4 \times 0,5 means 4 groups of 78844.png, which is 78839.png + 78834.png + 78830.png + 78824.png, which is 2.

0,5 \times 4 means 78817.png of 4, which is 2.

A real-life example where we would find this is:

6 \times 0,42 kg = 6 \times 78809.png 

= (6 \times 42) \div 100

= 252 \div 100

= 2,52 kg

What really happens is that we convert 6 \times 0,42 to the product of two whole numbers, do the calculation and then convert the answer to a decimal fraction again (\div 100).

multiplying decimals with whole numbers

1. Calculate each of the following. Use fraction notation to help you.

(a) 0,3 \times 7 (b) 0,21 \times 91 (c) 8 \times 0,4

= 82381.jpg \times 7

= 82388.jpg \times 91

= 8 \times 82396.jpg 

= 82353.jpg 

= 82362.jpg 

= 82370.jpg 







2. Estimate the answers to each of the following and then calculate:

(a) 0,4 \times 7 (b) 0,55 \times 7





(c) 12 \times 0,12 (d) 0,601 \times 2





3. Make a rule for multiplying with decimals. Explain your rule to a partner.




78496.png 

What does multiplying a decimal with a decimal mean?

For example, what does 0,32 \times 0,87 mean?

If you buy 0,32 m of ribbon and each metre costs R0,87, you can write it as 0,32 \times 0,87.

0,32 \times 0,87 = 87803.png \times 87795.png [Write as common fractions]

= 87788.png [Multiplication of two fractions]

= 87780.png [The product of the whole numbers 32 \times 87]

= 0,2784 [Convert to a decimal by dividing the product by 10 000]

The product of two decimals is thus converted to the product of whole numbers and then converted back to a decimal.

The product of two decimals and the product of two whole numbers with the same digits differ only in terms of the place value of the products, in other words the position of the decimal comma. It can also be determined by estimating and checking.

Multiplying decimals with decimals

1. Calculate each of the following. Use fraction notation to help you.

(a) 0,6 \times 0,4 (b) 0,06 \times 0,4 (c) 0,06 \times 0,04

= 82320.jpg \times 82312.jpg 

= 82332.jpg \times 82327.jpg 

= 82348.jpg \times 82339.jpg 

= 82285.jpg 

= 82296.jpg 

= 82304.jpg 







Mandla uses this method to multiply decimals with decimals:

0,84 \times 0,6 = (84 \div 100) \times (6 \div 10)

= (84 \times 6) \div (100 \times 10)

= 504 \div 1 000

= 0,504

2. Calculate the following using Mandla's method:

(a) 0,4 \times 0,7 (b) 0,4 \times 7 (c) 0,04 \times 0,7























7.9 Division and decimal fractions

Look carefully at the following three methods of calculation:

1. 0,6 \div 2 = 0,3 [6 tenths \div 2 = 3 tenths]

2. 12,4 \div 4 = 3,1 [(12 units + 4 tenths) \div 4]

= (12 units \div 4) + (4 tenths \div 4)

= 3 units + 1 tenth

= 3,1

3. 2,8 \div 5 = 28 tenths \div 5

= 25 tenths \div 5 and 3 tenths \div 5

= 5 tenths and (3 tenths \div 5) [3 tenths cannot be divided by 5]

= 5 tenths and (30 hundredths \div 5) [3 tenths = 30 hundredths]

= 5 tenths and 6 hundredths

= 0,56

dividing decimals by whole numbers

1. Complete.

(a) 8,4 \div 2

= (8


+ 4 tenths) \div 2

= (8


\div 2) + (


)

= 4


+


tenths

=


(b) 3,4 \div 4

= (3 units + 4 tenths) \div 4

= (32


+ 20


) \div 4

= (


\div 4) + (


\div 4)

=


+


hundredths

=


2. Calculate each of the following in the shortest possible way:

(a) 0,08 \div 4 (b) 14,4 \div 12





(c) 8,4 \div 7 (d) 4,5 \div 15





(e) 1,655 \div 5 (f) 0,225 \div 25





3. A grocer buys 15 kg of bananas for R99,90. What do the bananas cost per kilogram?



4. Given 26,8 \div 4 = 6,7. Write down the answers to the following without calculating:

(a) 268 \div 4 (b) 0,268 \div 4 (c) 26,8 \div 0,4







5. Given 128 \div 8 = 16. Write down the answers to the following without calculating:

(a) 12,8 \div 8 (b) 1,28 \div 8 (c) 1,28 \div 0,8







6. Sue pays R18,60 for 0,6 metres of material. What does one metre of material cost?


7. John buys 0,45 m of chain. The chain costs R20 per metre. What does John pay for the chain?


8. You may use a calculator for this question.

Anna buys a packet of mincemeat. It weighs 0,215 kg. The price for the mincemeat is R42,95 per kilogram. What does she pay for her packet of mincemeat? (Give a sensible answer.)



In this chapter you will learn about quantities that change, such as the height of a tree. As the tree grows, the height changes. A quantity that changes is called a variable quantity or just a variable. It is often the case that when one quantity changes, another quantity also changes. For example, as you make more and more calls on a phone, the total cost increases. In this case, we say there is a relationship between the amount of money you have to pay and the number of calls you make.

You will learn how to describe a relationship between two quantities in different ways.

8.1 Constant and variable quantities 203

8.2 Different ways to describe relationships 205

Maths%20A%20Grade%207%20Chapter%208%20frontispiece.tif 

8Relationships between variables

8.1 Constant and variable quantities

Look for connections between quantities

1. (a) How many fingers does a person who is 14 years old have?


(b) How many fingers does a person who is 41 years old have?


(c) Does the number of fingers on a person's hand depend on their age? Explain.



There are two quantities in the above situation: age and the number of fingers on a person's hand. The number of fingers remains the same, irrespective of a person's age. So we say the number of fingers is a constant quantity. However, age changes, or varies, so we say age is a variable quantity.

2. Now consider each situation below. For each situation, state whether one quantity influences the other. If it does, try to say how the one quantity will influence the other quantity. Also say whether there is a constant in the situation.

(a) The number of calls you make and the amount of airtime left on your cellphone





(b) The number of houses to be built and the number of bricks required





(c) The number of learners at a school and the duration of the mathematics period







If one variable quantity is influenced by another, we say there is a relationship between the two variables. It is sometimes possible to find out what value of the one quantity, in other words what number, is linked to a specific value of the other variable.

3. Consider the following arrangements:

67462.png 

(a) How many yellow squares are there if there is only one red square?


(b) How many yellow squares are there if there are two red squares?


(c) How many yellow squares are there if there are three red squares?


(d) Complete the flow diagram below by filling in the missing numbers.

Can you see the connection between the arrangements above and the flow diagram? We can also describe the relationship between the red and yellow squares in words.

67594.png 

In words:

The number of yellow squares is found by multiplying the number of red squares by 2 and then adding 2 to the answer.

Input numbers Output numbers

(Number of red squares) (Number of yellow squares)

(e) How many yellow squares will there be if there are 10 red squares?


(f) How many yellow squares will be there if there are 21 red squares?


8.2 Different ways to describe relationships

Complete some flow diagrams and tables of values

Arelationship between two quantities can be shown with a flow diagram. In a flow diagram we cannot show all the numbers, so we show only some.

1. Calculate the missing input and output numbers for the flow diagram below.

Each input number in a flow diagram has a corresponding output number. The first (top) input number corresponds to the first output number. The second input number corresponds to the second output number, and so on.

We say \times 2 is the operator.

(a)

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(b) What types of numbers are given as input numbers?


(c) In the above flow diagram, the output number 14 corresponds to the input number 7. Complete the following sentences in the same way:

In the relationship shown in the above flow diagram, the output number


corresponds to the input number 5.

The input number


corresponds to the output number 6.

If more places are added to the flow diagram, the input number


will correspond to the output number 40.

2. Complete this flow diagram by writing the appropriate operator, and then write the rule for completing this flow diagram in words.

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In words:

Multiply the input number by 4.


3. Complete the flow diagrams below. You have to find out what the operator for (b) is and fill it in yourself.

(a) (b)

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4. Complete the flow diagram:

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Acompleted flow diagram shows two kinds of information:

The flow diagram that you completed in question 4 shows the following information:

The relationship between the input and output numbers can also be shown in a table:

Input numbers

0

1

5

9

11

Output numbers

3

5

13

21

25

5. (a) Describe in words how the output numbers below can be calculated.

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(b) Use the table below to show which output numbers are connected to which input numbers in the above flow diagram.

Input number

10

20

30

40

50

Output number

4

8

12

16

20

(c) Fill in the appropriate operator and complete the flow diagram.

67713.png 

(d) The flow diagrams in question 5(a) and 5(c) have different operators, but they produce the same output values for the same input values. Explain.




6. The rule for converting temperature given in degrees Celsius to degrees Fahrenheit is given as: "Multiply the degrees Celsius by 1,8 and then add 32."

(a) Check whether the table below was completed correctly. If you find a mistake, correct it.

Temperature in degrees Celsius

0

5

20

32

100

Temperature in degrees Fahrenheit

32

41

68

89,6

212

(b) Complete the flow diagram to represent the information in (a).

67738.png 

7. Another rule for converting temperature given in degrees Celsius to degrees Fahrenheit is given as: "Multiply the degrees Celsius by 9, then divide the answer by 5 and then add 32 to the answer."

(a) Complete the flow diagram below.

67759.png 

(b) Why do you think the flow diagrams in questions 6(b) and 7(a) produce the same output numbers for the same input numbers, even though they have different operators?




(c) Will the flow diagram below give the same output values as the flow diagram in question 7(a)? Explain.

67781.png 



8. The rule for calculating the area of a square is: "Multiply the length of a side of the square by itself."

(a) Complete the table below.

Length of side

4

6

10

Area of square

64

144

(b) Complete the flow diagram to represent the information in the table.

67803.png 

9. (a) The pattern below shows stacks of building blocks. The number of blocks in each stack is dependent on the number of the stack.

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Stack 1 Stack 2 Stack 3

Complete the table below to represent the relationship between the number of blocks and the number of the stack.

Stack number

1

2

3

4

5

6

7

8

Number of blocks

1

8

(b) Describe in words how the output values can be calculated.



EXTENSION: Linking flow diagrams, tables of values, and rules

  • •

1. Complete the flow diagrams below.

(a) (b)

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(c) (d)

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(e) (f)

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2. Calculate the differences between the consecutive output numbers and compare them to the differences between the consecutive input numbers. Consider the operator of the flow diagram. What do you notice?

The difference between consecutive output numbers is equal to the multiplicative

operator. The multiplicative operator is the difference per input number.

3. Determine the rule to calculate the missing output numbers in this relationship and complete the table:

Input numbers

1

2

3

4

5

7

10

Output numbers

9

16

23

The difference between the consecutive output numbers is 7. The multiplicative

operator is 7. If you multiply 7 by 1, you need to add 2 to get 9. The rule is multiply

by 7 and add 2.


You will remember from Grade 6 that perimeter is the distance around the outermost border of something. Area is the size of a flat surface of something. In this chapter, you will learn to use different formulae to calculate the perimeter and area of squares, rectangles and triangles. You will solve problems using these formulae, and you will also learn how to convert between different units of area.

9.1 Perimeter of polygons 213

9.2 Perimeter formulae 214

9.3 Area and square units 215

9.4 Area of squares and rectangles 218

9.5 Area of triangles 224

Maths%20Grade%207%20Chapter%209%20frontispiece.tif

How big is it?

9 Perimeter and area of 2D shapes

9.1 Perimeter of polygons

The perimeter of a shape is the total distance around the shape, or the lengths of its sides added together. Perimeter (P) is measured in units such as millimetres (mm), centimetres (cm) and metres (m).

Measuring perimeters

1. (a) Use a compass and/or a ruler to measure the length of each side in figures A to C. Write the measurements in mm on each figure.

(b) Write down the perimeter of each figure.

A B C

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2. The following shapes consist of arrows that are equal in length.

(a) What is the perimeter of each shape in number of arrows?

(b) If each arrow is 30 mm long, what is the perimeter of each shape in mm?

A B C

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D E F G

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9.2 Perimeter formulae

If the sides of a square are all s units long:

Perimeter of square = s +s + s + s

= 4 \times s

or P = 4s

39350.png

If the length of a rectangle is l units and the breadth (width) is b units:

Perimeter of rectangle = l + l + b + b

= 2 \times l + 2 \times b

= 2l + 2b

or P = 2(l + b)

39361.png

A triangle has three sides, so:

Perimeter of triangle = s1 + s2 + s3

or P = s1 + s2 + s3

61081.png

Applying perimeter formulae

1. Calculate the perimeter of a square if the length of one of its sides is 17,5 cm.



2. One side of an equilateral triangle is 32 cm. Calculate the triangle's perimeter.



3. Calculate the length of one side of a square if the perimeter of the square is 7,2 m. (Hint: 4s = ? Therefore s = ?)




4. Two sides of a triangle are 2,5 cm each. Calculate the length of the third side if the triangle's perimeter is 6,4 cm.




5. A rectangle is 40 cm long and 25 cm wide. Calculate its perimeter.




6. Calculate the perimeter of a rectangle that is 2,4 m wide and 4 m long.



7. The perimeter of a rectangle is 8,88 m. How long is the rectangle if it is 1,2 m wide?



8. Do the necessary calculations in your exercise book in order to complete the table. (All the measurements refer to rectangles.)

Length

Breadth

Perimeter

(a)

74 mm

30 mm

(b)

25 mm

90 mm

(c)

1,125 cm

6,25 cm

(d)

5,5 cm

22 cm

(e)

7,5 m

3,8 m

(f)

2,5 m

12 m

39402.png 

9.3 Area and square units

The area of a shape is the size of the flat surface surrounded by the border (perimeter) of the shape.

Usually, area (A) is measured in square units, such as square millimetres (mm2), square centimetres (cm2) and square metres (m2).

Square units to measure area

1. Write down the area of figures A to E below by counting the square units. (Remember to add halves or smaller parts of squares.)

87886.png

A is


square units.

B is


square units.

C is


square units.

D is


square units.

E is


square units.

2. Each square in the grid below measures 1 cm2 (1 cm \times 1 cm).

(a) What is the area of the shape drawn on the grid?


(b) On the same grid, draw two shapes of your own. The shapes should have the same area, but different perimeters.

61748.png 

Conversion of units

The figure on the right shows a square with sides of 1 cm.The area of the square is one square centimetre (1 cm2).

How many squares of 1 mm by 1 mm (1 mm2) would fit into the 1 cm2 square?


Complete: 1 cm2 =


mm2

41925.png

To change cm2 to mm2:

1 cm2 = 1 cm \times 1 cm

= 10 mm \times 10 mm

= 100 mm2

Similarly, to change mm2 to cm2:

1 mm2 = 1 mm \times 1 mm

= 0,1 cm \times 0,1 cm

= 0,01 cm2

We can use the same method to convert between other square units too. Complete:

From m2 to cm2:

1 m2 = 1 m \times 1 m

=


cm \times


cm

=


cm2

From cm2 to m2:

1 cm2 = 1 cm \times 1 cm

= 0,01 m \times 0,01 m

=


m2

So, to convert between m2, cm2 and mm2 you do the following:

Do the necessary calculations in your exercise book. Then fill in your answers.

1. (a) 5 m2 =


cm2 (b) 5 cm2 =


mm2

(c) 20 cm2 =


m2 (d) 20 mm2 =


cm2

2. (a) 25 m2 =


cm2 (b) 240 000 cm2 =


m2

(c) 460,5 mm2 =


cm2 (d) 0,4 m2 =


cm2

(e) 12 100 cm2 =


m2 (f) 2,295 cm2 =


mm2

9.4 Area of squares and rectangles

Investigating the area of squares and rectangles

1. Each of the following four figures is divided into squares of equal size, namely 1 cm by 1 cm.

A B C

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D

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42800.png

(a) Give the area of each figure in square centimetres (cm2):

Area of A:


Area of B:


Area of C:


Area of D:


(b) Is there a shorter method to work out the area of each figure? Explain.


2. Figure BCDE is a rectangle and MNRS is a square.

42813.png 

(a) How many cm2 (1 cm \times 1 cm) would fit into rectangle BCDE?


(b) How many mm2 (1 mm \times 1 mm) would fit into rectangle BCDE?


(c) What is the area of square MNRS in cm2?


(d) What is the area of square MNRS in mm2?


3. Figure KLMN is a square with sides of 1 m.

(a) How many squares with sides of 1 cm would fit along the length of the square?


(b) How many squares with sides of 1 cm would fit along the breadth of the square?


46377.png

(c) How many squares (cm2) would therefore fit into the whole square?


(d) Complete: 1 m2 =


cm2

Aquick way of calculating the number of squares that would fit into a rectangle is to multiply the number of squares that would fit along its length by the number of squares that would fit along its breadth.

Formulae: Area of rectangles and squares

In the rectangle on the right:

Number of squares = Squares along the length \times Squares along the breadth

= 6 \times 4

= 24

46409.png 

From this we can deduce the following:

Area of rectangle = Length of rectangle \times Breadth of rectangle

A = l \times b

(where A is the area in square units, l is the length and b is the breadth)

Area of square = Length of side \times Length of side

A = l \times l

= l2

(where A is the area in square units, and l is the length of a side)

The units of the values used in the calculations must be the same. Remember:

Examples

1. Calculate the area of a rectangle with a length of 50 mm and a breadth of 3 cm. Give the answer in cm2.

Solution:

Area of rectangle = l \times b

= (50 \times 30) mm2 or A = (5 \times 3) cm2

= 1 500 mm2 or = 15 cm2

2. Calculate the area of a square bathroom tile with a side of 150 mm.

Solution:

Area of square tile = l \times l

= (150 \times 150) mm2

= 22 500 mm2

The area is therefore 22 500 mm2 (or 225 cm2).

3. Calculate the length of a rectangle if its area is 450 cm2 and its width is 150 mm.

Solution:

Area of rectangle = l \times b

450 = l \times 15

30 \times 15 = l \times 15 or 450 \div 15 = l

30 = l 30 = l

The length is therefore 30 cm (or 300 mm).

Applying the formulae

1. Calculate the area of each of the following shapes:

(a) a rectangle with sides of 12 cm and 9 cm




(b) a square with sides of 110 mm (answer in cm2)




(c) a rectangle with sides of 2,5 cm and 105 mm (answer in mm2)





(d) a rectangle with a length of 8 cm and a perimeter of 24 cm





2. A rugby field has a length of 100 m (goal post to goal post) and a breadth of 69 m.

(a) What is the area of the field (excluding the area behind the goal posts)?


(b) What would it cost to plant new grass on that area at a cost of R45/m2?


(c) Another unit for area is the hectare (ha). It is mainly used for measuring land. The size of 1 ha is the equivalent of 100 m \times 100 m. Is a rugby field greater or smaller than 1 ha? Explain your answer.



3. Do the necessary calculations in your exercise book in order to complete the table. (All the measurements refer to rectangles.)

Length

Breadth

Area

(a)

m

8 m

120 m2

(b)

120 mm

mm

60 cm2

(c)

3,5 m

4,3 m

m2

(d)

2,3 cm

cm

2,76 cm2

(e)

5,2 m

460 cm

m2

4. Figure A is a square with sides of 20 mm. It is cut as shown in A and the parts are combined to form figure B. Calculate the area of figure B.

A B

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5. Margie plants a vegetable patch measuring 12 m \times 8 m.

(a) What is the area of the vegetable patch?

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(b) She plants carrots on half of the patch, and tomatoes and potatoes on a quarter of the patch each. Calculate the area covered by each type of vegetable?



(c) How much will she pay to put fencing around the patch? The fencing costs R38/m.



6. Mr Allie has to tile a kitchen floor measuring 5 m \times 4 m. The blue tiles he uses each measure 40 cm \times 20 cm.

63035.png

(a) How many tiles does Mr Allie need?





(b) The tiles are sold in boxes containing 20 tiles. How many boxes should he buy?



Doubling a side and its effect on area

When a side of a square is doubled, will the area of the square also be doubled?

The size of each square making up the grid below is 1 cm \times 1 cm.

1. (a) For each square drawn on the grid, label the lengths of its sides.

(b) Write down the area of each square. (Write the answer inside the square.)

2. Notice that the second square in each pair of squares has a side length that is double the side length of the first square.

3. Compare the areas of the squares in each pair; then complete the following:When the side of a square is doubled, its area


66500.png 

9.5 Area of triangles

Heights and bases of a triangle

The height (h) of a triangle is a perpendicular line segment drawn from a vertex to its opposite side. The opposite side, which forms a right angle with the height, is called the base (b) of the triangle. Any triangle has three heights and three bases.

46903.png 

AD = height BD = height CD = height

BC = base AC = base AB = base

In a right-angled triangle, two sides are already at right angles:

46944.png 

DF = height EF = height FG = height

EF = base DF = base DE = base

Sometimes a base must be extended outside of the triangle in order to draw the perpendicular height. This is shown in the first and third triangles below. Note that the extended part does not form part of the base's measurement:

46962.png 

JM = height KM = height LM = height

KL = base JL = base JK = base

1. Draw any height in each of the following triangles. Label the height (h) and base (b) on each triangle.

2. Label another set of heights and bases on each triangle.

46982.png 

Formula: area of a triangle

ABCD is a rectangle with length = 5 cm and breadth = 3 cm. When A and C are joined, it creates two triangles that are equal in area: \triangle}ABC and \triangle}ADC.

47097.png 

Area of rectangle = l \times b

Area of \triangle}ABC (or \triangle}ADC) = 47127.png (Area of rectangle)

= 47134.png (l \times b)

In rectangle ABCD, AD is its length and CD is its breadth.

But look at \triangle}ADC. Can you see that AD is a base and CD is its height?

So instead of saying:

Area of \triangle}ADC or any other triangle = 47141.png (l \times b)

we say:

In the formula for the area of a triangle, b means ‘base' and not ‘breadth', and h means perpendicular height.

Area of a triangle = 47145.png (base \times height)

= 47149.png (b \times h)

Applying the area formula

1. Use the formula to calculate the areas of the following triangles: \triangle}ABC, \triangle}EFG, \triangle}JKL and \triangle}MNP.

47225.png 

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2. PQST is a rectangle in each case below. Calculate the area of \triangle}PQR each time.

(a) (b)

47260.png 

(c) R is the midpoint of QS.

47280.png 



3. In \triangle}ABC, the area is 42 m2, and the perpendicular height is 16 m. Find the length of the base.





1. Calculate the perimeter (P) and area (A) of the following figures:

47299.png 

P =


P =


P =


A =


A =


A =


2. Figure ABCD is a rectangle: AB = 3 cm, AD = 9 cm and TC = 4 cm.

47318.png 

(a) Calculate the perimeter of ABCD. (b) Calculate the area of ABCD.

(c) Calculate the area of \triangle}DTC. (d) Calculate the area of ABTD.

In this chapter, you will investigate the formulae we can use to calculate the area of the outer surfaces of cubes and rectangular prisms. Using nets of these 3D objects will help you to understand how we get to these formulae. You will then explore the formulae we can use to calculate the amount of space that solid cubes and rectangular prisms take up. The amount of space is known as their volume. You will then come to understand the difference between the volume and the capacity of cubes and rectangular prisms. You will also learn about the units that are used to calculate surface area, volume and capacity, and you will find out how to convert between different units of measurement.

10.1 Surface area of cubes and rectangular prisms 231

10.2 Volume of rectangular prisms and cubes 236

10.3 Converting between cubic units 240

10.4 Volume and capacity 244

Maths1_Gr7_ch10_frontispiece1.tif 

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10 Surface area and volume of

3D objects

10.1 Surface area of cubes and rectangular prisms

Investigating surface area

1. Follow the instructions below to make a paper cube.

Step 1: Cut off part of an A4 sheet so that you are left with a square.

47665.png 

Step 2: Cut the square into two equal halves.

47683.png 

Step 3: Fold each half square lengthwise down the middle to form two double-layered strips.

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Step 4: Fold each strip into four square sections, and put the two parts together to form a paper cube. Use sticky tape to keep it together.

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2. Number each face of the cube. How many faces does the cube have?


3. Measure the side length of one face of the cube.


4. Calculate the area of one face of the cube.


5. Add up the areas of all the faces of the cube.



47374.png 

The surface area of an object is the sum of the areas of all its faces (or outer surfaces).

A cube has six identical square faces. A die (plural: dice) is an example of a cube.

A rectangular prism also has six faces, but its faces can be squares and/or rectangles. A matchbox is an example of a rectangular prism.

Cube Rectangular prism

47738.png 

Using nets of rectangular prisms and cubes

It is sometimes easier to see all the faces of a rectangular prism or cube if we look at its net. A net of a prism is the figure obtained when cutting the prism along some of its edges, unfolding it and laying it flat.

1. Take a sheet of paper and wrap it around a matchbox so that it covers the whole box without going over the same place twice. Cut off extra bits of paper as necessary so that you have only the paper that covers each face of the matchbox.

2. Flatten the paper and draw lines where the paper has been folded. Your sheet might look like one of the following nets (there are also other possibilities):

47753.png 

3. Notice that there are six rectangles in the net, each matching a rectangular face of the matchbox. Point to the three pairs of identical rectangles in each net.

4. Use the measurements given to work out the surface area of the prism. (Add up the areas of each face.)

47769.png


5. Explain to a classmate why you think the following formula is or is not correct:

Surface area of a rectangular prism = 2(l \times b) + 2(l \times h) + 2(b \times h)

6. Here are three different nets of the same cube.

47785.png 

(a) Can you picture in your mind how the squares can fold up to make a cube?

(b) If the length of an edge of the cube is 1 cm, what is the area of one of its faces?


What then is the area of all its six faces?


(c) Explain to a classmate why you think the following formula is or is not correct:Surface area of a cube = 6(l \times l) = 6l2

(d) If the length of an edge of the cube above is 3 cm, what is the surface area of the cube?


Working out surface areas

1. Work out the surface areas of the following rectangular prisms and cubes.

A B

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C D

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2. The following two boxes are rectangular prisms. The boxes must be painted.

Box A Box B

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(a) Calculate the total surface area of box A and of box B.






(b) What will it cost to paint both boxes if the paint costs R1,34 per m2?





3. Two cartons, which are rectangular prisms, are glued together as shown. Calculate the surface area of this object. (Note which faces can be seen and which cannot.)

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4. This large plastic wall measures 3 m \times 0,5 m \times 1,5 m. It has to be painted for the Uyavula Literacy Project. The wall has three holes in it, labelled A, B and C, as shown. The holes go right through the wall. The measurements of the holes are in mm.

47987.png

(a) Calculate the area of the front and back surfaces that must be painted.

Remember from the previous chapter:

1 cm2 = 100 mm2

1 m2 = 10 000 cm2









(b) Calculate the area of the two side faces, as well as the top face.





(c) Calculate the total surface area of the wall, excluding the bottom and the inner surfaces where the holes are, because these will not be painted.





(d) What will it cost if the water-based paint costs R2,00 per m2?





10.2 Volume of rectangular prisms and cubes

2D shapes are flat and have only two dimensions, namely length (l) and breadth (b). 3D objects have three dimensions, namely length (l), breadth (b) and height (h). You can think of a dimension as a direction in space. Look at these examples:

2D shape: rectangle 3D object: rectangular prism

48244.png 

3D objects therefore take up space in a way that 2D shapes do not. We can measure the amount of space that 3D objects take up.

Every object in the real world is 3D. Even a sheet of paper is a 3D object. Its height is about 0,1 mm.

Cubes to measure amount of space

We can use cubes to measure the amount of space that an object takes up.

1. Identical toy building cubes were used to make the stacks shown below.

66578.png

A B C

D E F

(a) Which stack takes up the least space?


(b) Which stack takes up the most space?


(c) Order the stacks from the one that takes up the least space to the one that takes up the most space. (Write the letters of the stacks.)



The space (in all directions) occupied by a 3D object is called its volume.

2. The figure on the right shows a rectangular prism made from 36 cubes, each with an edge length of 1 cm. The prism thus has a volume of 36 cubic centimetres (36 cm3).

65570.png

(a) The stack is taken apart and all 36 cubes are stacked again to make a new rectangular prism with a base of four cubes (see A below.) How many layers of cubes will the new prism be? What is the height of the new prism?



A B

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(b) Repeat (a), but this time make a prism with a base of six cubes (see B above).



(c) Which one of the rectangular prisms in questions (a) and (b) takes up the most space in all directions? (Which one has the greatest volume?)



(d) What will be the volume of the prism in question (b) if there are 7 layers of cubes altogether?



(e) A prism is built with 48 cubes, each with an edge length of 1 cm. The base

consists of 8 layers. What is the height of the prism?



Formula to calculate volume

You can think about the volume of a rectangular prism in the following way:

Step 1: Measure the area of the bottom face (also called the base) of a rectangular prism. For the prism given here: A = l \times b = 6 \times 3 = 18 square units.

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Step 2: A layer of cubes, each 1 unit high, is placed on the flat base. The base now holds 18 cubes. It is 6 \times 3 \times 1 cubic units.

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Step 3: Three more layers of cubes are added so that there are 4 layers altogether. The prism's height (h) is 4 units. The volume of the prism is:

48527.png

V = (6 \times 3) \times 4

or V = Area of base \times number of layers

= (l \times b) \times h

Therefore:

Volume of a rectangular prism = Area of base \times height

= l \times b \times h

Volume of a cube = l \times l \times l (edges are all the same length)

= l3

Applying the formulae

1. Calculate the volume of these prisms and cubes.

A B

65954.png
65959.png

C D

65983.png
65987.png

2. Calculate the volume of prisms with the following measurements:

(a) l = 7 m, b = 6 m, h = 6 m (b) l = 55 cm, b = 10 cm, h = 20 cm

(c) Surface of base = 48 m2, h = 4 m (d) Surface of base = 16 mm2, h = 12 mm

3. Calculate the volume of cubes with the following edge lengths:

(a) 7 cm (b) 12 mm

4. Calculate the volume of the following square-based prisms:

(a) side of the base = 5 mm, h = 12 mm (b) side of the base = 11 m, h = 800 cm

5. The volume of a prism is 375 m3. What is the height of the prism if its length is 8 m and its breadth is 15 m?






48557.png 

10.3 Converting between cubic units

Cubic units to measure volume

This drawing shows a cube (A) with an edge length of1 m. Also shown is a small cube (B) with an edge length of 1 cm.

66640.png

How many small cubes can fit inside the large cube?

Total number of 1 cm3 cubes in 1 m3 = 100 \times 100 \times 100 = 1 000 000

∴ 1 m3 = 1 000 000 cm3

Work out how many mm3 are equal to 1 cm3:

1 cm3 = 1 cm \times 1 cm \times 1 cm

=


mm \times


mm \times


mm

=


mm3

Cubic units:

1 m3 = 1 000 000 cm3

(multiply by 1 000 000 to change m3 to cm3)

1 cm3 = 0,000001 m3

(divide by 1 000 000 to change cm3 to m3)

1 cm3 = 1 000 mm3

(multiply by 1 000 to change cm3 to mm3)

1 mm3 = 0,001 cm3

(divide by 1 000 to change mm3 to cm3)

Working with cubic units

1. Which unit, the cubic centimetre (cm3) or the cubic metre (m3), would be used to measure the volume of each of the following?

(a) a bar of soap


(b) a book


(c) a wooden rafter for a roof


(d) sand on a truck


(e) a rectangular concrete wall


(f) a die


(g) water in a swimming pool


(h) medicine in a syringe


2. Write the following volumes in cm3:

(a) 1 000 mm3


(b) 3 000 mm3


(c) 2 500 mm3


(d) 4 450 mm3


(e) 7 824 mm3


(f) 50 mm3


3. Write the following volumes in m3:

(a) 1 000 000 cm3


(b) 4 000 000 cm3


(c) 1 500 000 cm3


(d) 2 350 000 cm3


(e) 500 000 cm3


(f) 350 000 cm3


4. Write the following volumes in cm3:

(a) 2 000 mm3


(b) 4 120 mm3


(c) 1,5 m3


(d) 34 m3


(e) 50 000 mm3


(f) 2,23 m3


5. A rectangular hole has been dug for a children's swimming pool. It is 7 m long, 4 m wide and 1 m deep. What is the volume of earth that has been dug out?


6. Calculate the volume of wood in the plank shown below. Answer in cm3.

88054.png 




7. The drawing shows the base (viewed from below) of a stack built with 1 cm3cubes. The stack is 80 mm high everywhere.

48678.png

(a) What is the volume of the stack?



(b) Complete the following:

Volume of stack = area of base


8. Calculate the volume of each of the following rectangular prisms:

(a) length = 20 cm; breadth = 15 cm; height = 10 cm





(b) length = 130 mm; breadth = 10 cm; height = 5 mm





(c) length = 1 200 cm; breadth = 5,5 m; height = 3 m





(d) length = 1,2 m; breadth = 2,25 m; height = 4 m





(e) area of base = 300 cm2; height = 150 mm





(f) area of base = 12 m2; height = 2,25 m





10.4 Volume and capacity

The space inside a container is called the internal volume, or capacity, of the container. Capacity is often measured in units of millilitres (ml), litres (â„“) and kilolitres (kl). However, it can also be measured in cubic units.

Equivalent units for volume and capacity

If the contents of a 1 â„“ bottle are poured into a cube-shaped container with internal measurements of 10 cm \times 10 cm \times 10 cm, it will fill the container exactly. Thus:

48786.png

(10 cm \times 10 cm \times 10 cm) = 1 â„“

or 1 000 cm3= 1 â„“

Since 1 â„“ = 1 000 ml

1 000 cm3 = 1 000 ml [1 â„“ = 1 000 cm3]

∴ 1 cm3 = 1 ml [divide both sides by 1 000]

Since 1 kl = 1 000 â„“

= 1 000 \times (1 000 cm3) [1 â„“ = 1 000 cm3]

= 1 000 000 cm3

= 1 m3 [1 000 000 cm3 = 1 m3]

This means that an object with a volume of 1 cm3 will take up the same amount of space as 1 ml of water. Or an object with a volume of 1 m3 will take up the space of 1 kl of water.

The following diagram shows the conversions in another way:

\times 1 000 \times 1 000

48832.png
48833.png

1 ml ⟶ 1 ℓ (or 1 000 ml) ⟶ 1 kl (or 1 000 ℓ)

1 cm3 ⟶ 1 000 cm3 ⟶ 1 m3 (or 1 000 000 cm3)

48837.png
48838.png

\times 1 000 \times 1 000

Conversion is the changing of something into something else. In this case, it refers to changes between equivalent units of measurement.

From the diagram on the previous page, you can see that:

Remember these conversions:

1 ml = 1 cm3

1 kl = 1 m3

Volume and capacity calculations

1. Write the following volumes in ml:

(a) 2 000 cm3


(b) 250 cm3


(c) 1 â„“


(d) 4 â„“


(e) 2,5 â„“


(f) 6,85 â„“


(g) 0,5 â„“


(h) 0,5 cm3


2. Write the following volumes in kl:

(a) 2 000 â„“


(b) 2 500 â„“


(c) 5 m3


(d) 6 500 m3


(e) 3 000 000 cm3


(f) 1 423 000 cm3


(g) 20 â„“


(h) 2,5 â„“


3. A glass can hold up to 250 ml of water. What is the capacity of the glass:

(a) in ml?


(b) in cm3?


4. A vase is shaped like a rectangular prism. Its inside measurements are 15 cm \times 10 cm \times 20 cm. What is the capacity of the vase (in ml)?

48889.png



5. A liquid is poured from a full 2 â„“ bottle into a glass tank with inside measurements of 20 cm by 20 cm by 20 cm.

49000.png

(a) What is the volume of the liquid when it is in the bottle?


(b) What is the capacity of the bottle?


(c) What is the volume of the liquid after it is poured into the tank?


(d) What is the capacity of the tank?



(e) How high does the liquid go in the tank?




49016.png 

In question 5 above, you should have found the following:

Volume of liquid in tank = Volume of liquid in bottle

20 \times 20 \times h (liquid's height in tank) = 2 000 cm3

h = 49023.png

= 5 cm

Note: The capacity of the tank is 20 cm \times 20 cm \times 20 cm = 8 000 cm3 (8 â„“).The volume of liquid in the bottle is 2 000 cm3 (2 â„“).

1. Do the following unit conversions:

(a) 2 348 cm2 =

0,2348

m2 (b) 5,104 m2 =

51 040

cm2

(c) 1 m3 =

1

kl (d) 250 cm3 =

250

ml =

0,25

â„“

(e) 0,5 kl =

500

â„“ =

500 000

ml (f) 6,850 â„“ =

6 850

ml =

6 850

cm3

2. A rectangular prism measures 8 m \times 4 m \times 3 m. Calculate:

(a) its surface area (b) its volume

3. Aboy has 27 cubes, with edges of 20 mm. He uses these cubes to build one big cube.

(a) What is the volume of the cube if he uses all 27 small cubes?




(b) What is the edge length of the big cube?



(c) What is the surface area of the big cube?




4. A glass tank has the following inside measurements: length = 250 mm, breadth = 120 mm and height = 100 mm. Calculate the capacity of the tank:

(a) in cubic centimetres




(b) in millilitres



(c) in litres


5. Calculate the capacity of each of the following rectangular containers. The inside measurements have been given.

Length

Breadth

Height

Capacity

(a)

15 mm

8 mm

5 mm


cm3

(b)

2 m

50 cm

30 cm


â„“

(c)

3 m

2 m

1,5 m


kl




6. A water tank has a square base with internal edge lengths of 150 mm. What is the height of the tank when the maximum capacity of the tank is 11 250 cm3?





Revision 250

Assessment 259

Revision

You should not use a calculator for any of the questions in this chapter, unless you are told to use one. Do show all your steps of working.

fractions

1. Calculate the following:

(a) 3 85568.png + 2 85560.png (b) 4 85550.png - 3 85543.png 

= 5 85868.jpg or 85860.jpg + 85851.jpg = 85843.jpg 

= 85888.jpg - 85880.jpg 

= 6 85822.jpg = 6 85815.jpg 

= 85835.jpg = 85828.jpg 

(c) 85438.png - 85430.png

(give your answer asa mixed number)

(d) 2 85397.png \times 1 85389.png 

= 85791.jpg - 85783.jpg 

= 85808.jpg \times 85800.jpg 

= 85762.jpg = 1 85755.jpg 

= 2 85773.jpg

2. Three quarters of a number is 63. What is the number?

Let the number be x. \times x = 63 ∴ x = \times = 252 \div 3 = 84

3. Write down all the fractions in this list that are smaller than one eighth:

85258.png; 85249.png; 85240.png; 85232.png 

;

4. The Stone Hill Primary U13A soccer team had a good season, winning five sixths of its matches. If the team played 12 matches that season, how many were lost?

of 12 = \times 12 = 10 matches were won.



5. For each sequence below, write down whether it is increasing, decreasing, or neither:

(a) 85178.png; 85171.png; 85163.png


(b) 85129.png; 85121.png; 85113.png


(c) 85081.png; 85073.png; 85063.png


(d) 85030.png; 85021.png; 85011.png


6. In a survey of 80 Grade 7 learners, 60% felt that Justin Bieber was the best singer. How many learners think he is the best singer?

60% of 80 = 0,60 \times 80 = 48 learners (or \times = 48)

7. Moeketsi collected R450 of the total of R3 000 collected by his class for the ABC for Life charity. What percentage of the total did Moeketsi collect?

450 \div 3 000 = 0,15 (or \times = 15%)



8. BestWear had a sale on all its dresses. What was the percentage reduction on a dress that used to cost R600, but on sale was going for R480?


120 \div 600 = 0,20 (or \times = 20) The percentage reduction was 20%.

the decimal notation for fractions

1. Re-order the following numbers from smallest to largest:

(a) 0,04; 84934.png; 14%; 0,4%

0,4%; 0,04; 14%;

(b) 0,798; 0,789; 0,8; 0,79



2. What is the value of the 7 in 4,5678? Write your answer as a common fraction.

84916.jpg

3. Fill in the missing numbers in the boxes below.

(a) 7,99

84889.png
84887.png
84886.png
84888.png

8 (b) 9,123; 9,121;

; 9,117; …

4. Join all the pairs of numbers that multiply together to give 1. The first has been done for you. Note that you will not use all the numbers on the right-hand side.

84850.png 

5. Calculate the following:

(a) 5,673 - 3,597 (b) 4,85 \times 1,2 (c) 4,825 \div 5







6. A certain portion of the shapes below are shaded. Write each portion as a common fraction (in simplest form), decimal fraction and percentage.

(a) (b)

84476.png
84485.png

85738.jpg ;0,25; 25%

85747.jpg ;0,4; 40%

relationshipS BETWEEN VARIABLES

1. (a) Here is a number sequence: 1; 4; 10; 22;


;


The rule for creating the number sequence is "times 2, add 2". Write down the next two numbers in the number sequence.

(b) Here is another number sequence: 100; 50; 25; …

Write down, in words, the rule for creating this number sequence.



2. Use the given rule to calculate the missing values and/or determine the rule.

(a) (b)

88472.png
88480.png

(c) (d)

88499.png

9

88630.png
88638.png
88639.png
88645.png
88651.png
88508.png

add

88667.png

1

1

1

3

10

88659.png
88666.png
88714.png

3. (a) There is a simple relationship (multiply by …) between the y values and the x values in the table. Find it and then fill in the missing values.

x

0,1

0,3

0,6

2,5

3,2

4,1

y

4

12

24

100

128

164

(b) Write in words the rule that describes the relationship between the x values and the y values.



4. (a) There is a simple relationship (add …) between the x values and the y values in the table. Find the relationship and then fill in the missing values.

x

84088.png 

84080.png 

84072.png 

9 84064.png

84056.png 

14

84039.png 

y

84031.png 

2

84022.png 

10

15

84004.png 

(b) Write in words the rule by which the missing x and y values can be calculated.

Add to the x value.

5. The rule used to describe the relationship between the x values and y values in the table is "double the x and then subtract 2". Use the rule to find the missing values and fill them in.

x

4

8

12

15

22

51

y

6

14

22

28

42

100

perimeter and area of 2d shapes

1. (a) A rectangle has an area of 48 cm2 and a length of 8 cm. How wide is it?




(b) A different rectangle has an area of 72 cm2, and is twice as long as it is wide. Determine the dimensions of this rectangle.


x2 = 36 therefore x = 6



(c) A triangle has a base of 10 cm and an area of 20 cm2. What is the height of the triangle?

Area of the triangle = (base \times height) = 20 cm2




(d) What is the length of the side of a square that has an area of 144 cm2?


l = 12 cm

2. An equilateral triangle with sides of 8,4 cm and a square have the same perimeter. Determine the length of the side of the square.





3. Calculate the area of the shaded figures.

(a) DEFG is a rectangle. Dimensions of the sides are as indicated.

88128.png


Area of triangle = (4 \times 2) = 4



(b) ABCD is a rectangle. AB = 5 cm and FC = 2 cm. Give your answer in square millimetres. (You may use a calculator in this question.)

83932.png




Area of BCF = area of ADE = (FC \times BC)

Area of BCF = (2 cm \times 2 cm) = 2 cm2





4. The garden of Mr and Mrs Mbuli is shown below, not to scale. There is a hedge all around the garden, except for the 2 metre wide gate (from A to B). The shaded area is grassed (the rest has trees, shrubs etc.).

Garden Dream quoted the Mbulis R5 per square metre to mow their lawn and R10 per metre to trim their hedge. VAT is included in these prices. What was the total amount that Garden Dream quoted?

83914.png









surface area and volume of 3d objects

1. How many litres of water will a fish tank with inside measurements of 1,2 m \times 60 cm \times 70 cm hold, if it is filled to the brim?




2. A rectangular prism has a length of 4 cm, a width of 10 cm and a volume of 240 cm3. What is the height of the prism?




3. A rectangular prism has a certain volume. Which of the following will double the volume of the prism? Tick the correct answer(s).

88804.jpg Doubling all the dimensions

Doubling the length only

Doubling the length and the width, and halving the height

88752.jpg Doubling the length and halving the width and keeping the height unchanged

4. Look at the diagram below of a rectangular prism made out of 16 cubes.

83895.png 

Draw on the same grid two different rectangular prisms with the same volume as the one shown.

5. The total surface area of a cube is 150 cm2. Determine the volume of the cube.





6. The volume of a cube is 64 cm3. Determine the total surface area of the cube.





7. In order to save water when flushing the toilet, Mrs Patel added a solid brick to the cistern. The internal dimensions of the cistern are 30 cm \times 30 cm \times 10 cm, and the brick together with other internal mechanisms have a volume of 1 000 cm3.

(a) Calculate how many litres of water the cistern holds if the water fills up to 5 cm below the top of the cistern.




(b) Suppose the Patel family flush the toilet an average of 12 times a day. Use your calculator to determine how many kilolitres of water they will use by this means in one year.



8. Njabulo wishes to varnish the outside of a wooden chest that is in the shape of a rectangular prism. The bottom of the chest does not need to be varnished as it is on the ground. The chest is 1,5 m long, 50 cm wide and 80 cm high. Determine, in square metres, the total surface area that will need to be varnished.





9. The image on the right shows the net of a rectangular prism drawn on a grid. If each block on the grid is a square with a side length of 1 unit, calculate:

83814.png

(a) The total surface area of the prism



(b) The volume of the prism



Assessment

In this section, the numbers indicated in brackets at the end of a question indicate the number of marks that the question is worth. Use this information to help you determine how much working is needed.

The total number of marks allocated to the assessment is 60.

Note:

Do not use your calculator!

83787.png 

1. 83779.png is half of x. What is the value of x? (2)

2 \times =

2. The diagram alongside shows a square made up of blocks. Eight of these blocks have been shaded. Write, in its simplest form, the fraction of the square that is shaded. (2)

83752.png

=

3. Calculate the following:

(a) 2 83727.png \times 1 83720.png (b) 83713.png - 83704.png (6)

= 85715.jpg \times 85706.jpg 

= 85729.jpg - 85722.jpg

= 85690.jpg = 4

= 85698.jpg 

4. Mrs Baker has baked a cake. She has some ladies around for tea and they eat half the cake. Her son John eats a quarter of the rest of the cake. What fraction of the cake is left? (2)

\times =

- = There is of the cake left.

5. The price of petrol has risen from R8 per litre to R12 per litre over the past 2 years. Determine the percentage increase in the price. (2)



6. The Cupidos moved home. In the move, 5% of their crockery got broken. They have 57 pieces of crockery left (unbroken). How many pieces broke in the move? (2)





7. 83550.png = 0,0375; 83543.png = 0,0425; 83536.png = 0,0475

Using the above information, write down the decimal equivalents of the following fractions:

(a) 83528.png =


(b) 83495.png =


(c) 83462.png =


(3)

8. Multiply 56,76147 by 100 and round off your answer to two decimal places. (2)



9. Buti goes to the store and buys two cooldrinks at R7,50 each and three packets of chips at R5,95 each. If he pays with a R50 note, how much change should he get? (4)






10. Class 7A at Grace Primary School collects some money for 3 charities. If the total they collect is R823,80, and the money is allocated equally to each charity, how much will each charity receive? (2)




11. Use the given rule to calculate the missing values: (3)

86372.png 

(a) There is a simple relationship (add …) between the values of x and those of y. Find the relationship and then write down the missing values into the table. (2)

x

0,15

0,76

0,99

1,71

17,68

y

1,4

2,01

2,24

2,96

18,93

(b) Write in words the rule by which the missing x and y values can be calculated. (1)


12. The total area of the rectangle shown is 112 cm2. Determine the lengths of a and b.

83399.png

(3)


a = 112 cm2 \div 10 cm = 11,2 cm

a = b + 6,8 cm



13. Below is a rectangle, with dimensions as shown. A square has the same perimeteras the rectangle below. How long is the side of the square? (2)

83391.png



14. The diagram shows a rectangle divided into a triangle and a trapezium. Calculate the shaded area, giving your answer in mm2.

83382.png

(5)



Area of the triangle = (9 cm - 5,5 cm) \times 4 cm





15. The length and width of a rectangle is doubled.

(a) Tick the statement that is correct:

83372.jpg The perimeter of the rectangle stays the same.

The perimeter of the rectangle doubles.

83345.jpg The perimeter of the rectangle increases but it is not possible to say exactly by how much.

(b) Tick the statement that is correct:

83343.jpg The area of the rectangle stays the same.

83341.jpg The area of the rectangle doubles.

83339.jpg The area of the rectangle triples.

The area of the rectangle increases to 4 times what it was before.

(c) Explain your answer to part (b). (3)



16. A rectangular prism has a volume of 24 cm3. In the table below, write four possible dimensions that the prism may have. One possible combination has already been added. Note: do not consider, for example, a prism with length 6 cm, and height and width 2 cm to be different. (4)

Length

Width

Height

2 cm

2 cm

6 cm

3 cm

2 cm

4 cm

4 cm

6 cm

1 cm

24 cm

1 cm

1 cm

17. The inside of the boot of a car is in the shape of a rectangular prism, with length 1,2 m, width 70 cm and depth 40 cm. Determine the capacity of the boot in litres. (3)




18. The volume of a cube is 27 cm3. Determine the surface area of the cube. (3)




19. The length and breadth of a rectangular prism are both 4 cm, and its volume is 48 cm3. Determine the height of the prism. (2)



20. Consider this net:

83291.png 

(a) What is the name of the solid created if this net is folded?


(b) Which corner will A touch when the solid is created: B, C or D? (2)


Notes to myself

Notes to myself

Notes to myself